definition of signature of Hermitian matrix
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of Hermitian matrix.
- The reader knows a definition of eigenvalues of square matrix over ring.
Target Context
- The reader will have a definition of signature of Hermitian matrix.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( n\): \(\in \mathbb{N} \setminus \{0\}\)
\( M\): \(\in \{\text{ the } n \times n \text{ Hermitian matrices }\}\)
\( (\lambda_1, ..., \lambda_n)\): \(= \text{ the eigenvalues of } M \text{ including any duplicated roots }\)
\(*(p, n, z)\): \(p = \text{ the number of the positive eigenvalues including any duplicated roots }, n = \text{ the number of the negative eigenvalues including any duplicated roots }, z = \text{ the number of the zero eigenvalues including any duplicated roots }\)
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Conditions:
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2: Note
The definition is well-defined, because the eigenvalues of \(M\) are all real, as a well known fact.
There are indeed exactly \(n\) roots including any duplicated roots, as a well known fact.
So, \(n = p + n + z\).
